On quasisymmetric mappings in semimetric spaces

TL;DR

This paper extends the concept of quasisymmetric mappings from metric to semimetric spaces, analyzing their properties and establishing new inequalities and connections with weak similarities.

Contribution

It generalizes quasisymmetric mappings to semimetric spaces and derives new conditions and inequalities, including a generalized diameter ratio estimate.

Findings

Conditions for preserving triangle functions and Ptolemy's inequality

A new estimation for the ratio of diameters of images of subsets

Connections between quasisymmetric mappings and weak similarities

Abstract

The class of quasisymmetric mappings on the real axis was first introduced by A. Beurling and L. V. Ahlfors in 1956. In 1980 P. Tukia and J. V\"{a}is\"{a}l\"{a} considered these mappings between general metric spaces. In our paper we generalize the concept of quasisymmetric mappings to the case of general semimetric spaces and study some properties of these mappings. In particular, conditions under which quasisymmetric mappings preserve triangle functions, Ptolemy's inequality and the relation ``to lie between'' are found. Considering quasisymmetric mappings between semimetric spaces with different triangle functions we have found a new estimation for the ratio of diameters of two subsets, which are images of two bounded subsets. This result generalizes the well-known Tukia-V\"{a}is\"{a}l\"{a} inequality. Moreover, we study connections between quasisymmetric mappings and weak similarities which are a special class of mappings between semimetric spaces.